Some uses of homogeneous forcing David Asper o University of East - - PowerPoint PPT Presentation

Some uses of homogeneous forcing

David Asper´

• University of East Anglia

Torino, 27–05–2015

A forcing notion P is homogeneous iff for all p, p0 2 P there are q P p and q0 P p0 such that P q ⇠ = P q0 Standard fact: If P is a homogeneous forcing notion, then for all p, p0 2 P and all statements ' in the forcing language for P with parameters from the ground model, p P ' iff p0 P '

Hilbert’s programme revisited

Some local notation: Given a theory Σ and a sentence , in the language of set theory, is a Φ–consequence from Σ, denoted Σ `Φ , iff for every set–forcing P, if P forces every sentence in Σ, then P forces . Φ is for ‘forcing’. This definition of course makes sense for choices of Σ for which this can be expressed. For choices of Σ where its members have unbounded L´ evy complexity this might of course not be

• definable. Also, note that the definition makes sense also for

choices of Σ which are not even definable (as long as they are in V).

This gives a notion of logic | =Φ, possibly weaker than the logic | =GM of the generic multiverse. We use Φ–true, Φ–satisfiable, Φ–complete and so on, in the natural intended way. For example, a theory Σ is Φ–complete for a set ∆ of sentences if and

• nly if for every 2 ∆ at least one of Γ |

=Φ and Γ | =Φ ¬ holds. The usual (Woodin’s) definition of Ω–logic can be phrased in the above language, at least for (say) choices of Σ which are definable over !: Suppose Σ is definable over !. Then is an Ω–consequence of Σ if and only if the sentence “for all ordinals ↵, if Vα | = for every 2 Σ, then Vα | = ” is a Φ–truth (where

• f course the mention of Σ refers to the definition of Σ).

We may also define relativized versions ΦΓ of Φ–logic for definable classes Γ of posets. For example T is ΦΓ–complete for ∆ iff for every 2 ∆ it holds that either

• for every P 2 Γ, if P ' for every ' 2 T, then P , or
• for every P 2 Γ, if P ' for every ' 2 T, then P ¬.

Σ2 theories

For Σ2 theories, i.e., theories of the form (9↵)(Vα | = T) (equivalently, of the form (9)(H() | = T)) and Σ2 sentences , Φ–logic coincides with Ω–logic: T | =Φ iff T | =Ω

Woodin: If there is a proper class of Woodin and the Ω Conjecture is true, then:

1 the Pmax–axiom (⇤) is Ω–satisfiable (equiv., it can always

be obtained by set–forcing over any set–forcing extension). Hence, since (⇤) is Φ–complete for Th(H(!2)), if the Ω Conjecture is true under every large cardinal hypothesis, then (⇤) is an axiom which is

• compatible with all large cardinals,
• Ω–complete for Th(H(!2)), and
• which can always be set–forced after any set–forcing.

2 There is no Ω–satisfiable theory which is Ω–complete for

Th(H(+

0 )), where 0 is the least Woodin cardinal.

Woodin: If there is a proper class of Woodin and the Strong Ω Conjecture is true, then:

1 The Ω Conjecture is true. 2 All theories which are Ω–complete for Th((H(!2)) imply

¬ CH.

3 There is no Ω–satisfiable theory which is Ω–complete for

the Σ2

3 theory.

In “Incompatible Ω–complete theories”, JSL 2009, Koellner and Woodin contemplate the following very optimistic scenario: Could it be, in a large cardinal context, that the following holds? (i) The Ω Conjecture is false. (ii) There is a sequence of Ω–satisfiable Σ2 theories which are Ω–complete for the theory of larger and larger (all ?) reasonably specifiable initial segments of the universe. (iii) All these theories give the same theory of the relevant initial segments of the universe. Koellner and Woodin show that if (i) and (ii) hold, then (iii) has to fail (granting liberal use of large cardinals, as usual).

In “Incompatible Ω–complete theories”, JSL 2009, Koellner and Woodin contemplate the following very optimistic scenario: Could it be, in a large cardinal context, that the following holds? (i) The Ω Conjecture is false. (ii) There is a sequence of Ω–satisfiable Σ2 theories which are Ω–complete for the theory of larger and larger (all ?) reasonably specifiable initial segments of the universe. (iii) All these theories give the same theory of the relevant initial segments of the universe. Koellner and Woodin show that if (i) and (ii) hold, then (iii) has to fail (granting liberal use of large cardinals, as usual).

They show that if there is a Σ2 theory T which, modulo some large cardinal assumption LC, is Ω–satisfiable and Ω–complete for (say) Th(H()), for  = (2@0)+, then there are Σ2 theories T CH, T ¬ CH which, modulo slightly stronger large cardinal assumption LC0, are Ω–satisfiable and Ω–complete for Th(H(!2)) and such that

• T CH ` CH and
• T ¬ CH ` ¬ CH.

Proof proceeds by considering the theories that (essentially) say “I am a forcing extension of a model of T by Add(!1, 1)” (for T CH) “I am a forcing extension of a model of T by Add(!, !2)” (for T ¬ CH) The main points are:

• Add(!1, 1) and Add(!, !2) are definable over H() from no

parameters and homogeneous.

•  is large enough that all nice names for members of

H(!2) are in H(). CH, ¬ CH is clearly not the only pair they can deal with. A similar result can be proved for any Σ2 statement such that both and ¬ can be forced by some similarly nice forcing.

Proof proceeds by considering the theories that (essentially) say “I am a forcing extension of a model of T by Add(!1, 1)” (for T CH) “I am a forcing extension of a model of T by Add(!, !2)” (for T ¬ CH) The main points are:

• Add(!1, 1) and Add(!, !2) are definable over H() from no

parameters and homogeneous.

•  is large enough that all nice names for members of

H(!2) are in H(). CH, ¬ CH is clearly not the only pair they can deal with. A similar result can be proved for any Σ2 statement such that both and ¬ can be forced by some similarly nice forcing.

Down to H(ω2)

Consider the question: Question: Does the existence of an Ω–satisfiable Σ2–theory T which is Ω–complete for Th(H(!2)) imply the existence of another such theory incompatible with T? [Koellner–Woodin] does not address this question: their use of Add(!, !2) does address the problem of producing a theory implying ¬ CH, but Add(!1, 1) is not suitable for building a theory implying CH (in our context): If CH fails, then there are nice Add(!1, 1)–names for members of H(!2) which are not in H(!2). In fact Add(!1, 1) will collapse !2.

Down to H(ω2)

Consider the question: Question: Does the existence of an Ω–satisfiable Σ2–theory T which is Ω–complete for Th(H(!2)) imply the existence of another such theory incompatible with T? [Koellner–Woodin] does not address this question: their use of Add(!, !2) does address the problem of producing a theory implying ¬ CH, but Add(!1, 1) is not suitable for building a theory implying CH (in our context): If CH fails, then there are nice Add(!1, 1)–names for members of H(!2) which are not in H(!2). In fact Add(!1, 1) will collapse !2.

Addressing the question

Plan: Use [Koellner–Woodin]’s result in the following form: Theorem [Koellner–Woodin] Suppose there is a proper class of Woodin cardinals. Suppose ' is a Σ2 large cardinal property and is a Σ2 sentence such that T= ZFC + + “There is a proper class of Wodin cardinals” + “There is a proper class of '–cardinals” is Ω–complete for Th(H(!2)). Let P ✓ H(!2) be a forcing such that T Ω–implies that (1) P is definable over H(!2) (from no parameters). (2) P is homogeneous. (3) P preserves !1 and has the @2–c.c. (in particular every P–name for a member of H(!2) can be assumed to be in H(!2)).

Let TP be the sentence: There is (, N, G) such that

•  is an inaccessible cardinal,
• N |

= T,

• G is PN–generic over H(!2)N, and
• H(!2) = H(!2)N[G].

Then the sentence ZFC+TP +“There is a proper class of Wodin cardinals” + “There is a proper class of '–cardinals” is Ω–complete for Th(H(!2)).

A definable homogeneous version of the Hechler iteration

Goal: Want to force b > !1 by a forcing P ✓ H(!2) such that: (1) P is definable over H(!2) (from no parameters). (2) P is homogeneous. (3) P preserves !1 and has the @2–c.c. (in particular every P–name for a member of H(!2) can be assumed to be in H(!2)). (4) P forces b > !1.

First approximation: Consider the following Hechler iteration: P = Pω2, where (Pα)αω2 is such that for all ↵: (a) Conditions in Pα are finite functions p = ((sβ, Fβ) : 2 dom(p)) such that dom(p) ✓ ↵ and for all 2 dom(p):

• sβ 2 <ω!
• Fβ is a finite set of names (in H(!2)) ˙

f such that Pβ ˙ f 2 ω!.

(b) Given p0 = ((s0

α, F0 β) : 2 dom(p0)),

p1 = ((s1

β, F1 β) : 2 dom(p1)) 2 Pα, p1 extends p0 iff

dom(p0) ✓ dom(p1) and for all 2 dom(p0),

• s1

β extends s0 β,

• F0

β ✓ F1 β, and

• for every ˙

f 2 F0

β and every k 2 |s1 β| \ |s0 β|,

p1 Pβ ˙ f(k) < s1

β(k).

P = Pω2 forces b > !1, it preserves !1 and has the @2–c.c. (in fact it has the c.c.c.), and it is homogeneous. On the other hand P does not seem to be definable over H(!2): The reference, in the definition of Pβ, to arbitrary Pα–names, for ↵ < , blows up the complexity of the definition. It is not clear that even Pω is definable over H(!2). How to ensure definability?

P = Pω2 forces b > !1, it preserves !1 and has the @2–c.c. (in fact it has the c.c.c.), and it is homogeneous. On the other hand P does not seem to be definable over H(!2): The reference, in the definition of Pβ, to arbitrary Pα–names, for ↵ < , blows up the complexity of the definition. It is not clear that even Pω is definable over H(!2). How to ensure definability?

Mimicking forcing iterations by definable homogeneous forcing

Notation: Given functions p, p0 with ranges consisting of

• rdered pairs, p and p0 are compatible iff for all

x 2 dom(p) \ dom(p0), p(x) = (Y, ⌧) and p0(x) = (Y 0, ⌧) for some Y, Y 0 and ⌧. Given functions p, p0 with ranges consisting of ordered pairs, if p and p0 are compatible, then p ^ p0 denotes the function with dom(p ^ p0) = dom(p) [ dom(p0) such that

• (p ^ p0) dom(p) \ dom(p0) = p dom(p) \ dom(p0),
• (p ^ p0) dom(p0) \ dom(p) = p0 dom(p0) \ dom(p), and
• for all x 2 dom(p) \ dom(p0), if p(x) = (Y, ⌧) and

p0(x) = (Y 0, ⌧), then (p ^ p0)(x) = (Y [ Y 0, ⌧).

Let Pb>ω1 be the following poset: q 2 Pb>ω1 iff q is a finite function consisting of pairs ((↵, P, ˙ F), (X, )) where: (1) ↵ 2 !2 (2) P 2 H(!2) is a partial order consisting of finite sets of pairs ((, Q, ˙ G), (Y, ⌧)), where

• 2 !2
• Q, ˙

G, Y, 2 H(!2), ⌧ 2 <ω!, and such that

• for all p, p0 2 P, if p and p0 are compatible, then p ^ p0 is a

common extension of p and p0 in P.

(3) ˙ F 2 H(!2) is a P–name for an !1–sequence of members

• f ω!.

(4) X 2 [!1]<ω (5) 2 <ω!

Given q0, q1 2 Pb>ω1, q1 extends q0 iff (a) dom(q0) ✓ dom(q1) (b) For every (↵, P, F) 2 dom(q0), if q0((↵, P, F)) = (X0, 0) and q1((↵, P, F)) = (X1, 1), then

(i) X0 ✓ X1, (ii) 0 ✓ 1, and (iii) for all ⌫ 2 X0 and all n 2 dom(1) \ dom(0), q1 \ P 2 P and there is some s 2 |σ1|! such that q1 \ P P ˙ F(⌫) |1| = ˇ s and such that 1(n) > s(n)

Proposition

Pb>ω1 has the following properties.

1 It is definable over H(!2) from no parameters. 2 It is homogeneous. 3 It has precalibre @1 (i.e., every uncountable collection of

conditions includes an uncountable set X such that for every x 2 [X]<ω there is a common lower bound for all conditions in x).

4 It forces b > !1.

We immediately get the following.

Theorem

Suppose there is, under some sufficiently strong large cardinal assumption LC, a recursively enumerable Ω–satisfiable Σ2–theory T such that T is Ω–complete for Th(H(!2)). Then there are, under a slightly stronger large cardinal assumption LC0, Ω–satisfiable recursively enumerable Σ2–theories T b=ω1 and T b>ω1 such that

1 T b=ω1 and T b>ω1 are both Ω–complete for the theory of

H(!2).

2 T b=ω1 ` b = !1 3 T b>ω1 ` b > !1

Proof: By the above proposition together with an application of the Koellner–Woodin argument with Add(!, !2) (for T b=ω1) and with Pb>ω1 (for T b>ω1). ⇤

Another way to do this: Weak forms of Club Guessing and their negations

Club Guessing at !1 (CG): There is a ladder system (Cδ : 2 Lim(!1)) such that for every club C ✓ !1 there is some such that Cδ \ C is finite. Interval Hitting Principle (IHP): There is a ladder system (Cδ : 2 Lim(!1)) such that for every club C ✓ !1 there is some such that [Cδ(n), Cδ(n + 1)) \ C 6= ; for a tail of n < !. Here, (Cδ(n))n<ω is the strictly increasing enumeration of Cδ. (IHP is due to Kunen and is sometimes called Kunen’s Axiom.)

Consider the following forcing PIHP.

Definition

Conditions in PIHP are pairs q = (~ c, ~ D) with the following properties.

• ~

c = (cδ : 2 S) is a finite function with S ✓ Lim(!1) and such that for every 2 S, cδ ✓ ! ⇥ is a finite strictly increasing function.

• ~

D = (Dδ : 2 T) is such that T ✓ Lim(!1) is finite and for every , D is a finite set of cofinal subsets of of order type !. Given (~ c0, ~ D0), (~ c1, ~ D1) 2 PIHP, (~ c1, ~ D1)  (~ c0, ~ D0) iff: (1) dom(~ c0) ✓ dom(~ c1) and c0

δ ✓ c1 δ for every 2 dom(~

c0). (2) For every dom(~ D0) ✓ dom(~ D1) and every 2 dom(~ D0), D0

δ ✓ D1 δ.

(3) For every 2 dom(~ c0) and n, n + 1 2 dom(c1

δ ) \ dom(c0 δ ), if

2 dom(~ D0), then [c1

δ (n), c1 δ (n + 1)) \ D 6= ; for every

D 2 D0

δ.

Proposition

PIHP has the following properties.

1 It is definable over H(!2) from no parameters. 2 It is homogeneous. 3 It has the c.c.c. 4 It forces IHP.

Let B denote Baumgartner’s forcing for adding a club of !1 with finite conditions. Adding many Baumgartner clubs of !1: Given a set X of ordinals, there is a forcing, which I will denote by AddB(X), with the following properties. (1) For every AddB(X)–generic G and every ↵ 2 X one can naturally extract a Baumgartner club CG

α from G. Moreover,

CG

α 6= CG α0 for ↵ 6= ↵0 in X.

(2) AddB(X) is proper and has the @2–c.c. (3) For every partition (X0, X1) of X into nonempty pieces, AddB(X) ⇠ = AddB(X0) ⇥ AddB(X1). In particular, if G is AddB(X)–generic and ↵ 6= ↵0 are in X, then CG

α is

B–generic over V[CG

α0].

(4) AddB(X) is homogeneous. It follows from (1), (2) and (3) that if ot(X) !2, then AddB(X) forces ¬ IHP.

Let B denote Baumgartner’s forcing for adding a club of !1 with finite conditions. Adding many Baumgartner clubs of !1: Given a set X of ordinals, there is a forcing, which I will denote by AddB(X), with the following properties. (1) For every AddB(X)–generic G and every ↵ 2 X one can naturally extract a Baumgartner club CG

α from G. Moreover,

CG

α 6= CG α0 for ↵ 6= ↵0 in X.

(2) AddB(X) is proper and has the @2–c.c. (3) For every partition (X0, X1) of X into nonempty pieces, AddB(X) ⇠ = AddB(X0) ⇥ AddB(X1). In particular, if G is AddB(X)–generic and ↵ 6= ↵0 are in X, then CG

α is

B–generic over V[CG

α0].

(4) AddB(X) is homogeneous. It follows from (1), (2) and (3) that if ot(X) !2, then AddB(X) forces ¬ IHP.

Defininition: Let X be a set of ordinals. AddB(X) is the following forcing: Conditions in AddB(X) are pairs of the form p = (f, F) with the following properties. (1) f is a finite function with dom(f) ✓ X and such that f(↵) 2 B for every ↵ 2 dom(f). (2) F is a finite function with dom(F) ✓ !1 such that for every 2 dom(F),

(a) is a countable indecomposable ordinal, (b) F() is a countable subset of X, (c) 2 dom(f(↵)) and f(↵)() = for all ↵ 2 dom(f) \ F(), and (d) for every 0 2 dom(F ) and every ↵ 2 F(), rank(F(0), ↵) < .

Given (f0, F0), (f1, F1) 2 AddB(X), (f1, F1) extends (f0, F0) iff

• dom(f0) ✓ dom(f1) and f0(↵) ✓ f1(↵) for every ↵ 2 dom(f0),

and

• dom(F0) ✓ dom(F1) and F0() ✓ F1() for every

2 dom(F0).

We immediately get the following.

Theorem

Suppose there is, under some sufficiently strong large cardinal assumption LC, a recursively enumerable Ω–satisfiable Σ2–theory T such that T is Ω–complete for Th(H(!2)). Then there are, under a slightly stronger large cardinal assumption LC0, Ω–satisfiable recursively enumerable Σ2–theories T IHP and T ¬ IHP such that

1 T IHP and T ¬ IHP are both Ω–complete for the theory of

H(!2).

2 T IHP ` IHP 3 T ¬ IHP ` ¬ IHP

Proof: By the above together with an application of the Koellner–Woodin argument with PIHP (for T IHP) and with AddB(!2) (for T ¬ IHP). ⇤

Stronger results involving Club–Guessing

Weak Club Guessing (WCG): There is a ladder system (Cδ : 2 Lim(!1)) such that for every club C ✓ !1 there is some such that Cδ \ C is infinite. It seems there are also forcing notions P¬ WCG, PCG such that

1 P¬ WCG and PCG are definable over H(!2) from no

parameters.

2 P¬ WCG and PCG are homogeneous. 3 P¬ WCG and PCG are proper and have the @2–c.c. 4 P¬ WCG forces ¬ WCG and ¬ IHP. 5 PCG forces CG.

There is of course a corresponding corollary mentioning theories, Ω–complete for the theory of H(!2), T ¬ WCG +¬ IHP and T CG.

P¬ WCG and PCG

• ‘mimick’ forcing iterations (like in the definition of Pb>ω1)

and also

• incorporate side conditions (like in the definition of PIHP

and of AddB(X)). This is still work under construction.

P¬ WCG and PCG

• ‘mimick’ forcing iterations (like in the definition of Pb>ω1)

and also

• incorporate side conditions (like in the definition of PIHP

and of AddB(X)). This is still work under construction.

Many natural questions spring from here. For example:

• In the presence of some reasonable sufficiently strong

large cardinal axiom. Do all Ω–satisfiable recursive Σ2–theories which are Ω–complete for the theory of H(!2) imply the existence of a Suslin tree? (If yes, then of course the Pmax axiom (⇤) could not be Ω–satisfiable.)

• Do all Ω–satisfiable recursive Σ2–theories which are

Ω–complete for the theory of H(!2) imply ¬ CH?

Further advertising AddB(X): Collapsing exactly @3

AddB(X) has other interesting uses. Here is an example:

• U. Abraham proves the following in On forcing without the

continuum hypothesis, J. Symbolic Logic, vol. 48, 3 (1983), 658–661:

Theorem

(Abraham) (ZFC) There is a poset P collapsing !2 and preserving all other cardinals. Abraham’s forcing is built as follows: Let A ✓ !2 such that !L[A]

2

= !V

2 (and then of course !L[A] 1

= !V

1 ). Then

P = Add(!, !1) ⇤ Coll(!1, !2)LA][ ˙

G]

• P collapses !2 and has a dense subset of size @2.
• Preservation of !1: If G is Add(!, !1)–generic,

Coll(!1, !2)L[A][G] is –closed in L[A][G], but certainly not in general in V[G]. However, Coll(!1, !2)L[A][G] is –distribuitive in V[G]: Given a Coll(!1, !2)L[A][ ˙

G]–condition p and a

Coll(!1, !2)L[A][G]–name ˙ F in V[G] for a function ˙ F : ! ! Ord, we may find a condition p0  p in Coll(!1, !2)L[A][G] deciding all of ˙

• F. We use the Cohen reals

added by G in order to guide this construction (in V[G]).

Question

(in Abraham’s paper) Can this be extended to higher cardinals? In particular, is there, in ZFC, a forcing collapsing exactly @3?

Theorem

(ZFC) There is a poset P collapsing @3 and preserving all other cardinals.

Construction of P: There is a poset P0 of size @2 preserving cardinals and adding a partial ⇤ω1–sequence (Cα : ↵ 2 S) such that {↵ 2 S : cf(↵) = !1} is stationary. In V1 = VP0 we may then fix A ✓ !3 such that !L[A]

3

= !3 and such that for every cardinal ✓ > !3, the set of N H(✓) such that

• |N| = @1,
• N \ H(!3)L[A] 2 L[A], and
• N \ H(!3)L[A] is internally approachable in L[A]

is a stationary subset of [H(✓)]@1. Still in V1, let P1 = AddB(!1)L[A] ⇤ ˙ Q, where ˙ Q is, in L[A]AddB(ω1), a name for Coll(!2, !3)L[A][ ˙

G]. Our poset will be P = P0 ⇤ ˙

P1, where ˙ P1 is a P0–name for P1. ⇤ Question: Is there, in ZFC, a forcing notion collapsing @4 and preserving all other cardinals? What about for any  6= !1, !2, !3?

Relative definability

We all know that in ZFC one can prove the existence of such

• bjects as Hausdorff gaps, Aronszajn trees, partitions of !1 into

@1–many stationary sets, and so on. Many of these existence proofs proceed by a specific construction of the relevant type of

• bject, where this construction is definable from any given
• bject p satisfying a certain property P: One establishes in

ZFC the existence of some p such that P(p), and then one runs the relevant construction with any fixed p such that P(p) as a parameter. Typical example: If ~ C is a ladder system, then there is a recursive construction of a Countryman line definable from ~ C (Todorˇ cevi´ c). I will look next at the question: “if A is such that P(A), does there exists a B such that Q(B) and B is definable from A?” for various classical properties P(x), Q(x) of combinatorial flavour pertaining the structure H(!2).

Some positive results

Given two properties P(x), Q(x), I will say that P(x) has definability strength at least that of Q(x) over hH(!2), 2i if there is a formula '(x, y) such that Q({b 2 H(!2) : H(!2) | = '(A, b)}) for every A 2 H(!2) such that P(A).

Proposition

(ZF) The following properties have the same definability strength over hH(!2), 2i.

1 x is a ladder system 2 x is a simplified (!, 1)–morass 3 x is an special Aronszajn tree with a witness 4 x is a Countryman line with a witness 5 x is an indestructible gap with a witness

Sample proof: If p = (C, (Xn)n2ω) is a Countryman line with a witness, then there is a ladder system definable from p: Let ✓ = !2, and let A ✓ !1 be defined from p in H(!2) and such that Lθ(p) = Lθ(A) = Lθ[A]. If  = !1, then in Lθ[A], p = ((, C), (Xn)n2ω), where C is a linear order on  and (Xn)n2ω is a decomposition of  ⇥  into chains. But then necessarily  = !L[A]

1

: Lθ[A] can see that (, C) embeds neither what it thinks is !1, nor its converse, nor any uncountable set of reals (not difficult to verify and first observed by Galvin), and therefore it believes (correctly) that || = @1: Lθ[A] sees that any partition tree for (, C) has to be an Aronszajn tree on its !1, and therefore it sees also || = @1. But then  = !L✓[A]

1

. Now we can pick the <L✓[A]–first ladder system

• n !1 in Lθ[A].

⇤

Some negative results

Theorem

1 It is consistent that there is an Aronszajn tree T, an

(!1, !1)–gap (~ f,~ g) in (ω!, <⇤) and a partition ~ S of !1 into @1–many stationary sets such that no ladder system is definable from (T, (~ f,~ g), ~ S).

2 If there is an inaccessible cardinal, then the following holds

in a symmetric submodel of a forcing extension of V: There is an Aronszajn tree, an (!1, !1)–gap in (ω!, <⇤) and a partition of !1 into @1–many stationary sets but there is no ladder system on !1.

Proof of (1): Start with a model with an @2–Aronszajn tree T. Let ~ S = (Sν)ν<ω2 be any partition of !2 into stationary sets. Let P be c.c.c. forcing for adding (!1, !1)–gap, let G be P–generic and let (~ f,~ g) be the generic gap added by G.

Claim

Every c.c.c. forcing Q preserves the Aronszajnness of T. In particular, T is Aronszajn in V[G].

Proof.

Otherwise there is a Q–name ˙ b for a cofinal branch through T and a subtree T 0 ✓ T of height !2 with countable levels such that Q ˙ b ↵ 2 T 0

α for every ↵ < !2. But for every regular  and

< , every tree of height  with levels of size less than has a –branch, and so T 0, and therefore also T, has an !2–branch, which is a contradiction.

In V[G]Coll(ω, ω1), every Sν remains a stationary subset of !V[G]

2

= !V

2 , !1 = !V 2 , and (~

f,~ g) is still a gap: Suppose G0 is Coll(!, !1)–generic over V[G] and r is a real in V[G][G0]. Then r 2 V[G ↵][G0] for some ↵ < !2. But then r cannot split (~ f,~ g). Finally, in V[G]Coll(ω, ω1) there cannot be any ladder system on !1 = !V[G]

2

definable from (T, (~ f,~ g), ~ S). Otherwise, by homogeneity of the collapse this ladder system would be in V[G], which is impossible.

Proof of (2): Let  be an inaccessible cardinal such that there is a –Aronszajn tree T, let (Sν)ν<κ be a partition of  into stationary sets, and let G be generic for Pκ

κ. Our model W will

be the symmetric submodel of the extension of V[G] by Coll(!, <) generated by the names fixed by an automorphism

• f Coll(!, <) fixing Coll(!, <↵) for some ↵ < . In W, every ↵

is collapsed to ! and so !1 = , each Sν is clearly stationary, T remains Aronszajn (a cofinal branch through T in W would have to be in V[G][H] for a Coll(!, <↵)–generic H for some ↵ < ), and (~ f,~ g) remains unsplit (by the same proof as in the first part). Also, in W there is no ladder system on !1 as such an object would be in V[G][H] for some H as above, which is impossible. ⇤

Partitions of ω1 into stationary sets

Fact

1 (ZF + the club–filter on !1 is normal) If ~

C is a C–sequence, then there is a partition of !1 into @1–many stationary sets definable from ~ C.

2 (ZF + DC) If ~

r = (rα)α<ω1 is a one–to–one !1–sequence of reals, then there is a partition of !1 into @0–many stationary sets definable from ~ r. I don’t know if the normality of the club–filter is need in the first part and if DC is needed in the second part. In fact I don’t even know whether ZF alone implies that if ~ r is a one–to–one !1–sequence of reals, then there is a stationary and co–stationary subset of !1 definable from ~ r.

Theorem

Let  ! be a nonzero cardinal. The following theories are equiconsistent.

1 ZFC + There is a measurable cardinal. 2 ZFC + There is a partition (Si)i<λ of !1 into stationary sets

such that no partition of !1 into more than –many stationary sets is definable from (Si)i<λ.

Proof: Let  be measurable. By a classical result of Kunen–Paris we may assume that there are distinct normal measures Ui on  for i < . We may then find stationary subsets Si of , for i < , such that for all i⇤ < , i⇤ is the unique i < such that Si⇤ 2 Ui. We may assume that each Si consists of inaccessible cardinals. In VColl(ω, <κ), let ˙ P be a homogeneous forcing preserving the stationarity of all Si and adding a club C of  = !1, C ✓ S

i<λ Si, together with enumerations (X i α)α<κ of Ui for each

i < such that for all ↵ 2 C \ Si, ↵ 2 T

β<α X i β.

Let H be Coll(!, <) ⇤ ˙ P–generic over V, let C be the generic club of  added by ˙ P over V Coll(ω, <κ), and suppose, towards a contradiction, that there is a cardinal 0 > and a partition (Ai)i<λ0 of !V[H]

1

=  into stationary sets definable from (Si)i<λ. By homogeneity of Coll(!, <) ⇤ ˙ P, (Ai)i<λ0 2 V. There must then be some i⇤ < and two distinct i0, i1 < 0 such that both Ai0 \ Si⇤ and Ai1 \ Si⇤ are stationary in V[H]. There can be at most one ✏ 2 {0, 1} such that Ai✏ \ Si⇤ 2 Ui⇤. In that case it follows that a final segment of Ai✏ \ Si⇤ is contained in C. But then Ai1✏ \ Si⇤ is non-stationary, which is a contradiction. And if no Ai✏ \ Si⇤ is in Ui⇤, then of course no Ai✏ \ Si⇤ is stationary, which again is a contradiction.

For the other direction, suppose (Si)i<λ is a partition of !1 into stationary sets such that there is no partition of !1 into more than –many stationary sets definable from (Si)i<λ. Let A be a set of ordinals definable from, and coding (Si)i<λ. We show that  = !1 is measurable in the ZFC–model HOD(A).This is easy if is finite; in fact, in this case, for every i < , the club filter on !1 restricted to Si is, in HOD(A), a –complete ultrafilter on .

If = !, fix any i < and assume towards a contradiction that there is no stationary S ✓ Si in HOD(A) such that the club filter

• n  is an ultrafilter in HOD(A). Then we can define from A a

✓–maximal assignment (St : t 2 T) of stationary subsets of Si, for some tree T ✓ <κ2, such that St0 ✓ St for all t ✓ t0 in T, and with the property that for every t 2 T, if t is not a maximal node in T, then {tah0i, tah1i} ✓ T and {Stah0i, Stah1i} is a partition

• f St into stationary sets.

By ✓–maximality of (St : t 2 T) and the countable completeness of the nonstationary ideal it follows then that there is X ✓ T of size @1 definable from A such that {St : t 2 X} is a set of pairwise disjoint stationary sets, which contradicts our choice of (Si)i<ω and A. ⇤

It would be interesting to explore the possibilities for (other) large cardinal axioms to be equiconsistent with “ZFC+P(x) has definability strength strictly greater than Q(x)” for other natural pairs of properties P(x), Q(x).

Recall that, for a nonzero n 2 !, δ1

n denotes the supremum of

the lengths of all ∆1

n–pre-wellorderings of the reals. It is not

clear how to convert the proof of above theorem into a corresponding consistency result over ZF, but one can easily prove such results starting with a model of ZF + AD. For example, a classical well–known result of Solovay is that, under AD, the club filter on δ1

1 = !1 is an ultrafilter and

therefore !1 cannot be partitioned into 2 stationary sets. The following theorem generalises this.

Theorem

(ZF + AD) For every n < !, δ1

2n+1 is a successor cardinal and

regular and, letting  be such that + = δ1

2n+1, Coll(!, ) forces

that there is a partition of (δ1

2n+1)V = !1 into 2n+1 1 stationary

sets but no partition of !1 into more than 2n+1 1 stationary sets.